Recent content by TranscendArcu

Does nobody have any ideas? I was wondering if it were possible to confirm Wolfram's answer via induction, but expanding the resulting binomial coefficients fron the n-1 to the n case is proving to be fairly difficult. Any help is appreciated.

Find an expression that is identical to \sum_{k=0}^n \binom{3n}{3k} According to Wolfram, the correct solution to this is: \frac{1}{3} \left(2(-1)^n + 8^n\right) But I'm not sure which identities of the binomial coefficient I'm supposed to use to prove this. Can anyone give me some...

Homework Statement There is the suggestion to use an alternative estimate for the slope $$\beta$$ in the linear regression $$y_i=\alpha+\beta x_i+\epsilon_i$$which is formulated as $$B=(y_{max} -y_{min})/(x_{max} - x_{min})$$ Prove that such a formulation of $$B$$ is unbiased. Homework...

Yes, I think I'll have to add convergence hypotheses. In which case, the proof is immediate from Jensen's Inequality.

I don't think that's what I'm trying to show. I want to show a weak inequality, as opposed to a strict equality. Also, you seem to have assumed a uniform pdf (ie. the 1/N), which does not seem a fair assumption to me.

So I went and got this from Wikipedia: I wouldn't have much of a problem accepting ∞ ≥ ∞ if I thought that the expectation of such an RV even existed. Since the expectation diverges in my example, it seems meaningless to me to discuss comparisons of its nonexistent expected value.

Homework Statement Prove that if X is a positive-valued RV, then E(X^k) ≥ E(X)^k for all k≥1 The Attempt at a Solution Why do I feel like this is a counter-example: X = {1,2,4,8,16,...} (A positive-valued RV) m(X) = {1/2,1/4,1/16,1/32,...} (A distribution function that sums to one) Yet...

I'm just thinking out loud here, but might it be conceivable to write the following equation for the general case: \tilde{\beta}_1 = \hat{\beta}_1 + \Sigma_{j=2}^k (\hat{\beta}_j) \frac{\Sigma_{i=1}^n x_{1i}(x_ji - \bar{x}_j)}{\Sigma_{i=1}^n x_{1i}(x_1i - \bar{x}_1)} + \Sigma_{j=2} ^k...

Homework Statement The Attempt at a Solution So I was wondering whether or not, in an instance of n observations and k explanatory variables, where the following is an accurate statement: That is, the estimate of beta_1 found by only regressing y on x_1 is equal to the the true multiple...

Homework Statement Consider an elastic string of length L whose ends are held fixed. The string is set in motion with no initial velocity from an initial position u(x, 0) = f (x). Assume that the parameter alpha = 1. Find the displacement u(x,t) for the given initial position f(x) The Attempt...

So I thought I might have to write something of the form: Assume the solution can be written u(x,t) = X(x)T(t). Thus, by the heat equation u_t = a^2 u_{xx}, we wind up with two linear differential equations. Namely, X'' + qX = 0 and T' + a^2 q T = 0. Now I have to find which values of q make q...

Homework Statement The Attempt at a Solution So I know that I must have boundary conditions u(0,t) = 0 and ux(L,t) = 0. My textbook recommends reducing the given boundary conditions to homogeneous ones by subtracting the steady state solution. But, I thought these were already...

So I just tried to plot the first couple of terms in the series in MATLAB. I've attached the resulting graph in this post. As you can see, my answer isn't doing the right thing -- it's not correctly approximating the solution. Now, granted, this is only seven terms, but it doesn't seem to be...

This is the books answer: my answer will be f(x) = \frac{1}{2} + \sum_{n=1}^∞ ( \frac{(2)(-1)^{n+1}}{n\pi}) cos(\frac{n\pi x}{2}) But they seem to think that we're skipping some terms (hence the 2n - 1 things), but I don't have any terms zeroing so I don't need to make that adjustment.

Since it says to use a cosine series, I presume that the question intends for us to use an "even" function extension for the piecewise. Thus, the values for f(x) = 0 between -2<x<-1 and f(x) = 1 for -1 < x < 0. Assigning f(x)'s extension in such a way makes the function even and thus permits us...